Differential Equations Partial Differential Equations and Boundary-Value Problems Syllabus

What is a singular Sturm-Liouville problem?

One where the regularity conditions fail: e.g. $r(x)=0$ at an endpoint, or the interval is infinite, or $p$ or $r$ is unbounded. Boundedness of $y$ at the singular endpoint typically replaces a separated boundary condition.

What is an eigenfunction expansion of a function $f$ relative to a Sturm-Liouville system?

$$f(x)=\sum_{n=1}^{\infty}c_{n}y_{n}(x),\quad c_{n}=\frac{\int_{a}^{b}p(x)f(x)y_{n}(x)\,dx}{\int_{a}^{b}p(x)y_{n}^{2}(x)\,dx}.$$ Fourier sine/cosine series are special cases.

Give the general linear second-order PDE in two variables and the discriminant used to classify it.

For $A u_{xx}+B u_{xy}+C u_{yy}+D u_{x}+E u_{y}+F u=G$, the discriminant is $B^{2}-4AC$.

State the classification of second-order linear PDEs by the discriminant $B^{2}-4AC$.

$B^{2}-4AC<0$: elliptic (e.g. Laplace's equation); $B^{2}-4AC=0$: parabolic (e.g. heat equation); $B^{2}-4AC>0$: hyperbolic (e.g. wave equation).

Give a canonical example PDE for each of the three classifications.

Elliptic: $u_{xx}+u_{yy}=0$ (Laplace). Parabolic: $u_{t}=k\,u_{xx}$ (heat). Hyperbolic: $u_{tt}=a^{2}u_{xx}$ (wave).

What is a homogeneous linear PDE and how does the superposition principle apply?

A linear PDE $L[u]=0$ (no forcing term) is homogeneous. Superposition: if $u_{1},\dots,u_{k}$ are solutions, then any linear combination $u=\sum_{i}c_{i}u_{i}$ is also a solution; this extends to infinite series of product solutions.

Outline the method of separation of variables for a PDE in $u(x,t)$.

Assume $u(x,t)=X(x)T(t)$, substitute into the PDE, and separate so each side depends on only one variable, equal to a constant $\pm\lambda$. This yields two ODEs; apply the homogeneous boundary conditions to the $X$-equation to get an eigenvalue problem, then combine solutions by superposition.

In separation of variables, why is the separation constant usually written as $-\lambda$, and how is its sign determined?

Writing it as $-\lambda$ makes the spatial ODE a standard eigenvalue problem $X''+\lambda X=0$. The homogeneous boundary conditions rule out $\lambda\le0$ (which give only the trivial solution), forcing $\lambda>0$ and a discrete spectrum of positive eigenvalues.

State the one-dimensional heat (diffusion) equation and the meaning of its constant.

$$\frac{\partial u}{\partial t}=k\frac{\partial^{2}u}{\partial x^{2}},$$ where $u(x,t)$ is temperature and $k=\frac{K}{\rho\gamma}>0$ is the thermal diffusivity ($K$ thermal conductivity, $\rho$ density, $\gamma$ specific heat).

On what two physical laws is the derivation of the heat equation based?

Conservation of energy (heat) within a rod element, and Fourier's law of heat conduction, which states heat flux is proportional to the negative temperature gradient, $q=-K\,u_{x}$.

Solve the heat equation $u_{t}=k u_{xx}$ on $0<x<L$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=f(x)$.

$$u(x,t)=\sum_{n=1}^{\infty}c_{n}\sin\frac{n\pi x}{L}\,e^{-k(n\pi/L)^{2}t},\quad c_{n}=\frac{2}{L}\int_{0}^{L}f(x)\sin\frac{n\pi x}{L}\,dx.$$

For the heat equation with insulated ends, $u_{x}(0,t)=u_{x}(L,t)=0$, what form does the solution take?

$$u(x,t)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos\frac{n\pi x}{L}\,e^{-k(n\pi/L)^{2}t},$$ a Fourier cosine series; as $t\to\infty$ the temperature approaches the average $\frac{a_{0}}{2}=\frac{1}{L}\int_{0}^{L}f(x)\,dx$.

How are nonhomogeneous boundary conditions for the heat equation handled by separation of variables?

Split $u(x,t)=\psi(x)+v(x,t)$ where $\psi(x)$ is the steady-state solution satisfying the nonhomogeneous BCs (from $\psi''=0$), and $v$ solves the heat equation with homogeneous BCs and adjusted initial condition $v(x,0)=f(x)-\psi(x)$.

For a rod with ends held at temperatures $u(0,t)=u_{0}$ and $u(L,t)=u_{1}$, what is the steady-state temperature $\psi(x)$?

Solving $\psi''(x)=0$ with $\psi(0)=u_{0}$, $\psi(L)=u_{1}$ gives the linear profile $$\psi(x)=u_{0}+\frac{u_{1}-u_{0}}{L}\,x.$$

State the one-dimensional wave equation for a vibrating string and the meaning of $a$.

$$\frac{\partial^{2}u}{\partial t^{2}}=a^{2}\frac{\partial^{2}u}{\partial x^{2}},$$ where $u(x,t)$ is transverse displacement and $a^{2}=\frac{T}{\rho}$ ($T$ = tension, $\rho$ = linear mass density), so $a$ is the wave speed.

What initial and boundary data are needed for the vibrating string fixed at both ends?

Boundary conditions $u(0,t)=u(L,t)=0$; and two initial conditions — initial displacement $u(x,0)=f(x)$ and initial velocity $u_{t}(x,0)=g(x)$ (the wave equation is second order in $t$).

Give the series solution of the wave equation on $0<x<L$ with fixed ends, $u(x,0)=f(x)$, $u_{t}(x,0)=g(x)$.

$$u(x,t)=\sum_{n=1}^{\infty}\left(A_{n}\cos\frac{n\pi a t}{L}+B_{n}\sin\frac{n\pi a t}{L}\right)\sin\frac{n\pi x}{L},$$ with $A_{n}=\frac{2}{L}\int_{0}^{L}f\sin\frac{n\pi x}{L}dx$ and $B_{n}=\frac{2}{n\pi a}\int_{0}^{L}g\sin\frac{n\pi x}{L}dx$.

State d'Alembert's solution of the wave equation $u_{tt}=a^{2}u_{xx}$ on the infinite line.

With $u(x,0)=f(x)$, $u_{t}(x,0)=g(x)$: $$u(x,t)=\tfrac{1}{2}\left[f(x-at)+f(x+at)\right]+\frac{1}{2a}\int_{x-at}^{x+at}g(s)\,ds.$$

Interpret d'Alembert's solution physically in terms of traveling waves.

The solution is a superposition of two waves: $f(x-at)$ travels to the right and $f(x+at)$ travels to the left, both with speed $a$ and unchanging shape. The integral term accounts for the influence of the initial velocity.

What is the domain of dependence of the point $(x,t)$ in d'Alembert's solution?

The interval $[x-at,\;x+at]$ on the initial line $t=0$. Only initial data within this interval affect $u(x,t)$, reflecting finite propagation speed $a$.

What are the normal modes of a vibrating string fixed at both ends?

The individual product solutions $$u_{n}(x,t)=\left(A_{n}\cos\frac{n\pi a t}{L}+B_{n}\sin\frac{n\pi a t}{L}\right)\sin\frac{n\pi x}{L}.$$ Each mode oscillates with a fixed spatial shape $\sin\frac{n\pi x}{L}$; the general vibration is their superposition.

Give the natural (angular) frequency and nodes of the $n$-th normal mode of a string of length $L$.

Angular frequency $\omega_{n}=\frac{n\pi a}{L}$ (frequency $f_{n}=\frac{na}{2L}$). The $n$-th mode has $n-1$ interior nodes (fixed points where $\sin\frac{n\pi x}{L}=0$) besides the two endpoints.

What is a standing wave, and how does it relate to normal modes?

A standing wave is an oscillation with a fixed spatial pattern of nodes and antinodes that does not travel; each normal mode $\sin\frac{n\pi x}{L}\cos(\omega_{n}t)$ is a standing wave formed by the superposition of two oppositely-traveling waves of equal amplitude and frequency.

State Laplace's equation in two dimensions and what problems it models.

$$\nabla^{2}u=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0.$$ It models steady-state (time-independent) phenomena: steady temperature distributions, electrostatic potentials, and incompressible irrotational flow.

What are Dirichlet and Neumann boundary conditions for Laplace's equation?

Dirichlet: the value of $u$ is prescribed on the boundary. Neumann: the normal derivative $\frac{\partial u}{\partial n}$ is prescribed on the boundary. A mixed (Robin) condition prescribes a combination $\alpha u+\beta\frac{\partial u}{\partial n}$.

Give the separated solution of Laplace's equation on a rectangle $0<x<a$, $0<y<b$ with $u=0$ on the two vertical sides, $u(x,0)=0$, $u(x,b)=f(x)$.

$$u(x,y)=\sum_{n=1}^{\infty}A_{n}\sin\frac{n\pi x}{a}\sinh\frac{n\pi y}{a},\quad A_{n}\sinh\frac{n\pi b}{a}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{n\pi x}{a}\,dx.$$

What is the maximum principle for solutions of Laplace's equation?

A nonconstant harmonic function on a bounded region attains its maximum and minimum values only on the boundary of the region, never at an interior point.

Write Laplace's equation in polar coordinates $(r,\theta)$.

$$\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}}=0.$$

Give the general series solution of Laplace's equation inside a disk of radius $c$ with $u(c,\theta)=f(\theta)$.

$$u(r,\theta)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}r^{n}\left(a_{n}\cos n\theta+b_{n}\sin n\theta\right),$$ where $c^{n}a_{n},c^{n}b_{n}$ are the Fourier coefficients of $f(\theta)$; terms $r^{-n}$ are dropped for boundedness at $r=0$.

When Laplace's or the heat equation is separated in cylindrical (polar) coordinates, what ordinary differential equation arises in the radial variable, and what are its solutions?

The radial equation is Bessel's equation of order $n$, $$r^{2}R''+rR'+(\lambda^{2}r^{2}-n^{2})R=0,$$ whose solutions are the Bessel functions $J_{n}(\lambda r)$ and $Y_{n}(\lambda r)$; $Y_{n}$ is discarded when boundedness at $r=0$ is required.